Views:17803|Rating:4.81|View Time:4:21Minutes|Likes:173|Dislikes:7 With the rapid growth of cities around the world, global infrastructure demands are on the rise, which means the demand for civil engineers with advanced skills in on the rise, too.
A master’s degree in civil engineering will allow you to gain the advanced knowlege in your engineering specialization that will lead to greater career prospects (not to mention, salary!) and the skills to engineer better structures and systems as society creates more complex infrastructure to keep up with rapid growth.
what's up everybody Isaac here with civil engineering Academy jumping on giving you another quick tip today we're going to talk about the value of getting a master's degree and whether that is worth your time as a civil engineer to get so hang out with me today and we'll answer that question okay so the question again is as a master's degree worth it in civil engineering and today I'm going to talk about the benefits of getting a master's degree I've often kind of preached that getting your PE is looked at more valuable than getting a master's but obviously a master's degree is going to be really valuable as well in fact some people move up in a company don't even have their PE license but they're able to get into management and other positions without their PE even though they have a master's degree so let's talk about some of the benefits first of all the US Department of Labor says that the median salary for a civil engineer is about $85,000 that was in twenty twenty seventeen so that's a pretty good starting salary I think well not a starting salary but a median salary you know what I mean so eighty eighty-five thousand dollars now if you have your master's degree that actually usually is a little bit more than that so a master's degree that's a benefit right off the bat is that gaining a master's you also are paid a couple thousand dollars more to have that degree the other thing that's a benefit of having a master's degree is that it really specializes you so if you go into school you're gonna take classes that you enjoy if you want to go deeper into and it gives you an edge on the competition that is out there so if you're a structural engineer you're gonna take additional concrete courses structural loading courses still courses anything do a structural structures or structure of structural material so all those things and getting an edge with more education more education always gives you an edge on your career so getting a master's degree also helps with that the other benefit of getting a master's degree is that also future proofs yourself so by having a master's degree you are set up for the future right and what I mean by that is that you don't need to go back to school to get if you can knock out your master's degree while you're in school earning your bachelor's right after that would be my recommendation the other thing that's coming up is the ncees organization has been talking about for years trying to get people to have a master's degree or an additional 30 credit hours under their belt so they can become professional engineers now that's ten additional courses which is a master's equivalent and if you already have your master's degree you're already set up for that so you don't need to worry about it so having a master's degree really future proofs yourself you don't need to worry about coming back to school to get those things if you don't have your PE license or whatnot so anyway guys that's the benefits of getting a master's degree I do highly recommend it there are plenty opportunities out there is whether it's online through a school all your coursework online or if you actually attend a classroom so I've seen plenty of good programs with either one I've actually done a master's online as well and found it very helpful with time management especially have kids or family so anyway I do think a master's is worth it so get out there and find one that works for you and I think it will benefit you in your career alright guys that'll wrap it up we'll see you next time bye
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Views:1917|Rating:4.55|View Time:14:29Minutes|Likes:10|Dislikes:1 This lecture, held at American University of Sharjah (AUS) on Monday, February 10, 2014, viewed the achievements of medieval Islamic mathematics from the point of view of two questions: “What was (and wasn’t) Islamic about it?” and “What does the term `mathematics’ mean when used to describe the results of its practitioners?”. The AUS lecture presented evidence suggesting that it is only in the context of the Islamic faith interacting with the mathematical heritage from older civilizations that we can gain a full picture of mathematics in medieval Islam. In the course of providing some answers to our questions, this AUS lecture looked at some examples of Islamic achievements, both practical and theoretical, in such areas as arithmetic, algebra and geometry.
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my title of my talk is mathematics in medieval Islam on its own terms and I want to talk to you about the kinds of things we can learn when we try when we approach the mathematics of medieval Islam on its own terms and not on terms that we choose it has been sometimes in the past a tendency to treat mathematics in medieval Islam as if its principle achievement was to preserve the mathematics of the Greeks and pass it on to the eager minds of the European Renaissance and that has been more or less the said the debt we owe to medieval Islam now we all know I'm sure everyone in this room knows and many people in the West know that we got you know the hindu-arabic numerals so Islam was responsible for bringing us the hindu-arabic numerals and usually that story ends there about numeration in medieval Islam but the story is much more interesting and much richer than that when you begin to try to understand the mathematics in its own terms which is the theme of my talk and the three different kinds of calculation one was the arithmetic of the hand hisab a Yad which was not counting on your fingers but it was using mental arithmetic to carry on computations and intermediate results were sort of stored on these storage registers here on your on your hands and the fingers were held in certain positions to indicate numbers up to a thousand okay so you might write the number out the whole number out in the Hindi numerals but then when you came to fractions it's switch bases from 10 to 60 and you treat it as if it was degrees minutes and seconds and so he and eben Taher Alberico daddy in the 11th century when he wrote his compendium on reckoning included hand reckoning and the base 60 system along with the decimal system of the Hindus in fact not only did reckoning in can include these three systems it also included algebra and the the very eminent professor at Harvard Hameed Sabra wrote in his article on his Saab in the Encyclopaedia of Islam that many books are which were called his sab were in fact books on algebra and you can kind of understand why that might be because after all what is arithmetic about it's computing things and welcome you know what you're computing or you don't know what you're computing in other words the unknown it's still a matter of computing and so that's I think how you know algebra became at least in the medieval terms part of his sad so it was really only until about oh the I would say about the 14th century that the hisab al yad began to drop out and you don't find it mentioned in the books and in fact some of the theorems that one will find there are this kind of thing the identities for the signs of sums and differences of angles which would be a great aid in constructing trigonometric tables the law of sines which was both for plane and spherical triangles and that much simplified the task of surveyors and astronomers in calculating unknown parts of triangles from the knowing parts of the triangles and the third methods of computing trigonometric tables to arbitrary accuracy and in fact it's interesting to look at the the accuracy to which these tables were Commun computed as the as the years went on around the Year 900 Albertini gave a table in his book on astronomy of the sine function at intervals of half a degree to three places of accuracy 130 years later Albie Rooney I gave tables of the sine and tangent functions to steps of a quarter of a degree for the sine function and 1 degree for the tangent function and 2 for places of accuracy and then by 1440 Aloka beg the the king or the ruler the Prince of Samarkand computed these tables of the sine steps of 1 minute okay one minute 60 and a degree 90 degrees to a quadrant okay 16 times 90 5450 400 computations right carried out to five places of accuracy so these are these are the remarkable computational achievements and it's it seems almost incredible that people could have done this when there was no there were no really computing devices of the sort that were used to and but he didn't earn his living as a mathematician that's what I'm saying al carro G who discovered the binomial theorem in algebra al-qura G was an engineer Nasser Adina – see I had mentioned who invented trigonometry I was an astronomer Jamshid Alka she was an astronomer most of these people were supported by the wealthy either the the rulers or prominent families or undertook particular jobs in the cases of engineers and so on so this is you just can go through the list of people who are mathematicians or we'd say we're mathematicians and you'd find really they were earning their living doing something else teachers that was one way you could earn a living working in mathematics you could become a tutor there was not a lot of mathematics taught at the at the basic school level the idea there was that you would learn the elements of Arabic so that you could read the Quran properly and understand what you were reading and there was also the idea that you could you could go on to the madrasa and there were mattresses and they were a very intelligent faculty and very learned that people but they were focused more on the religious sciences than on the foreign sciences as as as the sciences kept aend from the Greeks and the Persians and the Hindus and so on were called and so there was not a lot of formal instruction but if you wanted to learn advanced mathematics you hired a tutor and tutors in fact were were common way of learning advanced mathematics in fact one of the one of the things that came out of the teaching of mathematics and this was especially in the Maghreb was the introduction of mathematical symbols in writing mathematical texts so for example machine men for shine the thing the unknown okay or meme for mal property or it came to mean the square of the unknown calf for cab okay the the cube okay Jim for zither the root the square root okay and League for equals Moussaoui Li and so the lamb was simply for the equal sign so this developed out of the pedagogical environment in the in the Maghreb especially in Morocco but in in nearby countries there are two the mathematicians tried to answer some of these charges in various ways I mean the idea was show that mathematics at an advanced level was important to the Muslim community and so even Khaldoon remarks in his MOOC Adama he says religious talking about the law of inheritance which of course one does involve mathematics I mean going back to al-khwarizmi in the ninth century his book on algebra includes a number of inheritance problems and he shows how to use algebra to solve the inheritance problems so but even the code Dylan says religious scholars in the Muslim cities have paid much attention to it the laws of inheritance some authors are inclined to exaggerate the mathematical side and to pose problems requiring for their solution various branches of arithmetic such as algebra the use of roots and similar things it is of no practical use in inheritance matters because it deals with unusual and rare cases so now there were also cartographic solutions to religious problems such as finding the direction of Mecca one of these right does the three things I want to discuss the new wounds the prayers and the direction of Mecca this was an interesting early map here it's a rectangular projection it represents some of the cities of the of the Muslim world I think you can see here Mecca okay and there they're saying that okay take a city such as Isfahan and then just draw a line joining Isfahan de Mecca and read off the angle to the Meridian and that will give you the direction of Mecca from Isfahan well it would if the earth were flat but the earth isn't flat it's certainly a reasonable approximation if you're somewhere not too far from Mecca ok but once you get into Central Asia and Samarkand and so on this is not going to be accurate but that was the illustration you see here for Mecca Isfahan and it says ok there's the number of degrees for the Qibla now there were much more sophisticated solutions than this that were developed and I just want to show you another one here this was a remarkable device this is one of those devices that if it had not been found nobody would believe that it existed it was made in the 17th century in Iran it is suspected a David King suspects that in fact it actually goes back much earlier than this to the 10th century in Baghdad but that's that's his speculation on this so what it is is the following you've got the you've got lines of longitude here you have lines of latitude here and in each little square I think these are two degrees steps of longitude and latitude okay there is written the name of some city who is located at that longitude and latitude and to find the Qibla what you do is you rotate this ruler until the edge passes through the particular city where you live okay and then you can read off from the degrees there you can read off what is the Qibla of Mecca and this is accurate it requires no computation or whatever it requires no tables whatever the work is all done in the mathematical projection of the longitude and latitude lines and now as I say it never rains it pours since they found the first one they found two more the third one was found sitting in the basement at Harvard University nobody knew it was there but these these devices were developed some personally develop it was a genius I mean this mathematical projection it's quite remarkable I think anyway what I mean by looking at medieval Islamic mathematics in its own terms I will hope you will find something within the lecture both in terms of so what those terms were and what the achievements of the resulting mathematics was that justifies your presence here today and thank you for your attention
Views:16247|Rating:4.61|View Time:1:42Minutes|Likes:164|Dislikes:14 In this video, you will get information about where and how to find civil engineering lectures of MIT college i.e. Massachusetts Institute of Technology, USA.
1) You can find lectures (in written format) here:
2) Link to MIT’s YouTube channel (for videos):
mit doesn't have much of the video lectures available on civil engineering however you can download PDF format of their lectures along with assignments to download civil engineering lectures from MIT you can simply google civil engineering lectures MIT then you can see the link over here oka mit.edu with title civil engineering and environmental engineering you can click the link and open it in new tab as page opens you can see the course and the lectures as you scroll down under various civil and environmental engineering related topics covering both undergraduate and graduate studies can simply open or click on any of these topics and download the lecture or you can simply type ocw.mit.edu are of your internet browser and visit their page then select department within find courses in the menu after new page loads you can click on civil and environmental engineering within School of Engineering section whatsoever to watch their videos you can subscribe their channel on YouTube called MIT OpenCourseWare and search for civil engineering videos there the link is given in the video info section below thank you for watching our video please do like and subscribe our Channel
Views:7|Rating:0.00|View Time:58:13Minutes|Likes:0|Dislikes:0 SUNY Provost and Vice Chancellor Tod Laursen lectures students and faculty members in the University at Buffalo’s Department of Civil, Structural and Environmental Engineering. His lecture is titled “Mortar-Based Approaches to Contact Problems in Computational Solid and Structural Mechanics.
marcelina senior vice chancellor for most of history university of system in September 2018 before coming to SUNY he was the president of the Valley Fire University in Abu Dhabi United Arab Emirates he was the founding president and sole its leader since 2010 and then before prior to becoming president of the Khalifa University it was a member of the faculty at the University and he served in the chair of course earned his PhD and master's degrees in mechanical engineering from Stanford University and he specializes in computational mechanics in the subfield of engineering mechanics concerned with development of new computational algorithms and tools used by engineers he has published over 100 referee in articles book chapters and after abstracts and his authored or co-authored two books his particular focus is development of methods to analyze contact impact and friction of phenomena in highly nonlinear and complex systems he's a fellow of the American Society of Mechanical Engineers the International Association of computational mechanics and the United States Association for computational mechanics she also holds membership and how the pie and he served as an at-large member of the executive committee for the United States Association for computational mechanics between 2007 and 2010 and currently services as they as a member of the Executive Council of the International Association for computational mechanics Anto 2020 and additionally he has served on the scientific advisory committees of several shape most important national national congress and computational mechanics small trishul faculty it's really a pleasure to be presenting a few slides with the equations on them because they don't get to somehow do this quite as we kind of went back and forth we met at a faculty meeting a few weeks ago very fine and extended the invitation to give a lecture in all today and iterating a little bit you mentioned the the two books and in one in particular that's sort of a research monograph things over the last 20 25 years we've thought about in computational mechanics and for contact problems in particular so in the way the lecture I'm going to give today is going to sort of pick up where the book leaves off in a way and you know I think the overriding goal of our work I'm a fine an element person by background and nonlinear continuum mechanics I would say and when I sort of entered this field as graduate student a situation we had and I was working in the National Laboratories in trying the situation we have we have fairly robust methods for things like computational plasticity computational geomechanics cat models for countries the kinds of things that those of you who you know work in these media I think will recognize we sort of knew and they were coming out I would say in the 80s and the well mostly in the 80s and to a certain extent in the late 70s these methods were pretty well standardized provided elements and how to treat those sorts of problems but for contact mechanics the story was a little bit different so you showed me nano intentions right in your laboratory so my first research project as graduate student was funded by IBM and it was sort of the early indentation studies that were done by IBM and folks at Oak Ridge and folks like this having to do with characterization of the thin films that that are put on disks right both both the magnetic media but then the protective coatings that are put over those and so so the question in those days and I know there are people on your faculty who are still grappling with that question is to what extent can we use continuum mechanics to describe things that are getting to let's say 20 to 30 nanometers something like this so this was my challenge as a graduate student to try to sort of use methods of continuum mechanics and computational mechanics to try to study these problems but you can't get around contact you can't get around friction and so this was part of my original for a if you like into thinking about this topic that was the application that I had in mind but I'll show you a few others and I'm actually going to use some slides that were provided by abacus a few years ago further applications they like to sell using contact mechanics and I think it gives you at least some kind of snapshot as to what the current state of the art is in this field the mortar methods which is really the need of my talk today several collaborators over a 10 or 15 year period some of the graduate students some of the collaborators in the national lab martinandheike in particular and then the other thing I'm just a little bit about is fluid structure interaction and how some of the things that we thought about in doing contact mechanics solid to solid or structure to structure sort of have some interesting potential applications at leaves for FSI and so just to give you a little bit of an idea again from a numerical point of view how one goes about doing so the rarity said a lot of this but so the idea here keep it in mind I'm sort of a methods developer okay the idea here is to sort of come up with tools you can think of them hopefully as sort of being under the hood not anything if you're a user of a code like Atticus you have to worry about but what we were really after was use whatever a friction model you want to use whatever interface model you want to use just like you employ a plasticity model or an elasticity model or a anisotropy model for a bulk solid we'd like to give you the same sort of robustness if you have to describe contact interactions but it's quite a bit different right and so I've got a sort of an embarrassingly old movie here but if you sort of watch this thing go so this is a tread block simulation we did permission one a few years ago so we'll finally – for me basically the sub linear is one solid actually contacting the roadway that you can't see here it's just sort of taken as a rigid you know transparent obstacle right that's actually the easy part of this problem the harder part and you can just sort of see some hint of it as this thing goes he's the contact that occurs inside the sipes and the grooves as this thing rolls and these are the kinds of things that if you tend to run your tires too long which I tend to do you can actually see those wear patterns and so the interest michelin had was to actually model that and see if we couldn't get at the energy dissipation that was happening there and again you can think of your favorite application you've got contact you've got friction you've got large deformations you've got potentially large relative motions and so we'd like to come up with methodologies that will work sort of without limitation on those things and then we also are not going to talk about it today there's also some interesting things that come up when you're doing true dynamics right and you want to get transient phenomena a lot of the same concerns come into play so I'm going to sort of use and I'll go through them pretty quickly and then I started to use this set of slides that my colleagues at abacus gave me a few years ago just to sort of give you some ideas from the sort of tire rolling industry and again you should visualize your favorite application just to sort of give you an idea of some of the things people would like to do with these methods and what's now possible so I should do our silly sort of academic tire that we were sort of doing to demonstrate concepts for from Michelin but in fact you know if you're using industrial software these days you can do these things pretty much in real time abacus is pretty well connected with a lot of major tire manufacturers and so people can look at two-dimensional you know axisymmetric models they can spin those or they can develop full 3d models and literally roll those down the roadway there are some interesting things that can come into play here you can just roll this thing down the road and try to sort of understand the dissipation it's occurring within the tire you can theoretically simulate things like skidding or cornering operations all of these things are possible you can look in detail at the sort of slip profiles over a contact patch and by the way another feature of contact problems we were talking about a little bit in our meeting earlier is that oftentimes if you've got an interface you think of taking a tire pressing it down into a roadway and then rolling it so the vast majority of that footprint is steep it's not and that's a good thing right especially when it's snowing but but actually in a normal sort of steady state rolling operation there's what's called kick-out region and so if this thing is rolling down the road you'll have a highly localised band this is just being pushed it's not rolling on one side that you actually want to resolve because since it's slipping there there's friction there right and so that has to do with energy dissipation and it's something we'd like to understand so so these are the kinds of things that these guys think about it's actually possible to couple them with system dynamics models and so you can start to view some things also like coupling a tire with a brake system experiment with different control schemes for for example yes type of braking right and then look at what that sort of makes happen in terms of the system dynamics and so they can look at things like the clamping force that's necessary again look at energy dissipation which is going to relate directly to things like like brake pad light and in a manner that's completely coupled with what the tire is doing on the right through the dynamic system all this gets at something I'm going to talk about at the very end at very end of the time so fluid structure interaction I know there's folks here actually in civil engineering who are doing important FSI work wind engineering has to think about that right so it's a very common event example but you actually see it in the tire business quite a bit too and so folks would like to understand if you drive through a puddle for example what is the loss of traction can we actually simulate that one of the things these guys have done think about you've had a tire you got a roadway and you've got intervening fluid because it's a contact problem you don't actually know a priori and any instant you know where the contact is going to be active you know where the pressure and the fluid is actually going to be positive and it's actually helping to hold the tire up and in what reasons you you sort of got a free surface none of this you know you also if you think about it for a minute and you think it's sort of a little grungy and meshing sort of way really don't want to have to fix a grid on the fluid and have the grid the form of the fluid it would actually be better think about how you usually do CFD I've taken our Larian framework for the fluid and let the solid work through it okay and so there are some methods and I'm going to talk a little bit later about how we actually do this where you have the capability to roll this tire for example through a puddle and it's not only notice how this grid here if the fluid does not conform to the physical boundary so there are ways to take care of those boundary conditions properly and what they've also figured out how to do you can't really see it very well here but to add adaptively reform other you know we grid this region so that you get resolution of the dynamics that are happening there so it's just sort of one example but this is the kind of thing that folks are interested in being able to do there's also the whole topic of particular particulate dynamics and uncapped right so southern the tire industry this could happen with snow obviously this could happen with gravel and so here we've got contact between you know continued and mechanically described solid but then effectively particle dynamics and so having robust methods to do like things like that is also an interest okay so so again let me just sort of give sort of a wish list things that we'd like to do here remembering we do numerical measure so I'd like to be able to tell you something if you're gonna use my tool about how accurate it is and how at what rate you know the error will reduce did you refined anything first of all you'd like to know is convergent right you'd like to know that if you use a finer grid you'll get a closer answer so we'd like to be able to tell you something about that accuracy the robustness is something that I'll allude to in a couple of points but maybe the way to keep in mind here and let me just kind of turn this movie loose so you can see it keep in mind with all of these problems that we don't know a priori we're going to put a boundary condition on the cylinder and sort of drag it across this rubber solid but we don't know how priori its knots on the boundary condition right we don't know at any particular time where the contact is going to establish where it's going to be free we don't know where the interface money stick or where it might slip we want the prediction tool to tell us that the trouble is that it's out of contact it's in contact you'll see graphs in a minute it's non smooth right most numerical methods and I don't know how many of you in here do numerical methods but think back to the first time you saw a find a different method or something right we're always relying on something like a Taylor series expansion one way or the other to sort of derive a discrete method that we can use so what do they tell you about Taylor series expansions you're not supposed to apply them to things that aren't smooth right so this isn't smooth this isn't smooth because I'm out of contacts and I'm in contact it also isn't smooth because I'm sticking and then I'm slipping and it happens abruptly right so coming up with something that's robust that's reasonably accurate that you could use in two or three dimensions all of those are are sort of challenges and I'm going to tell you a little bit about how we do it so let me just give you an idea of what the equations are again just sort of an intuitive idea of what we're talking about and prior to the work that we did here you know the wave was usually taken to do these problems and then I want to talk to you about water based methods which is what we found out for the last 10 or 15 years or so and then like I said I'll give a couple of extensions to related problems at the end so you know we're usin large deformations primarily so by and large there'll be some examples later that don't follow this formula but by and large we used Lagrangian methods which basically means that your mesh that you put on the solid is going to deform with that solid in time okay so we have some preference unloaded configurations and configuration you can notice automatically you can do it as a user specification but the idea here is you want to pick out all of the points on the surface of either of these bodies and make sure that they don't interpenetrate the other body that's sort of the idea in terms of impenetrability and so you've got your usual equilibrium F is equal to Ma in the continuum mechanical form you can have displacement and force or traction boundary conditions and we allow all of that if you've got dynamics you've got the appropriate initial conditions but then you have to have these constraints and these constraints are the ones that aren't smooth okay so we were to draw a picture of them this first one is sort of the out of contact in contact idea right so if you've studied if you're an Operations research person or if you studied constrained optimization problems one way to interpret mechanically what we're dealing with here is a classical Lagrangian supplier problem right so what what happens with a little rod multiplier problem you've got sort of a sine function it's defined for each point from the sources either in these bodies or both and we're going to define that function usually we define it so it's the distance between the point and the surface that it's opposing with a sign on okay and the physical idea here is simply that that point has to stay outside the surface to the other body can't go in okay so that's the G less than or equal to zero this TN is in mechanics we call it contact pressure but mechanically or I should say mathematically that's little brunch multiplier that's going to enforce that constraint now the thing that's a little bit different here this is a continuum mechanical description right so that Lagrange multiplier at least in theory could be defined for every point on the outside of either of those bodies and we have to figure out the regions where we require a contact pressure to be developed to keep those things from interpenetrating okay so that's the idea and then what we're gonna do here if you studied plasticity theory for the friction law is something quite similar to sort of a rigid plastic type of idealization the only difference is that rather than having a yield stress like you'd have in rigid plasticity sort of our yield stress if you want is going to be mu times the pressure okay and twist here so in other words if that confining pressure gets bigger then typically speaking with most interface is the sort of frictional yield stress if you want to think about it that way also increases so so we've got this here and you have a different sort of variety of optimization techniques to try to sort of solve for those multipliers so skipping a lot of details and we take this system and we dis precise it which means we put a finite element national on it you end up with a structural dynamics equation set that looks probably pretty familiar to most right so you've got your inertial terms if we were dealing with a linear system then app internal would just be K times D okay but it's basically the forces generated under the nodes by the stresses inside the bodies we'll talk about this guy in a moment and these are just your applied loads on the right okay so this would just sort of be your standard structural dynamic system if you were doing structural dynamics sort of a civil engineering sense you might have a damping matrix times DDOT in there too right but a sort of tricky guide for us is this contact force vector and again without getting into too much detail those are the forces required on nodes to keep those bodies from inter penetrating with each other okay so the question is we have very standard finite element recipes or how to go from say can we take the internal stress so if I take you know this which is basically div Sigma is equal AB right you can apply sort of a standard finite element recipe to come up with those internal forces in the you know in a finite element sense what the mortar method is going to be about eventually is how to do the same thing to turn these constraints which are again point-wise fields over the surfaces of both bodies how to convert those into what I'm calling FC here okay so it's a spatial discretization problem and I'm going to kind of walk you through you know how we figured out a way to do that that gives us some of these characteristics we said you want now if I go back to the time when I was a graduate student this is how everybody does contact and the first contact algorithms I row look like this so you know imagine you've got these two bodies you've put grids on them right now you're trying to decide sort of heuristic Li where do I enforce this constraint that nodes don't go inside the other body right well sort of a natural choice to make is just to pick all of the nodes on one side of the interface or the other side or maybe both and enforce that constraint in a discrete way point wise it doesn't necessarily seemed unreasonable right so a mathematician would call that a collocation approach you've got some condition that you're wanting to enforce and you have to pick a set of points where you're going to enforce it sort of the easiest way to do that is just say well I'll do it at the nose right it seems reasonable most of the algorithms we actually use for a lot of contact problems are based on it it's actually not right at least it's not right for the most challenging problems I know sort of showed you why that is in fact I've got a few quick examples right now that sort of give an idea of that so this is actually out of money dissertation so this is getting quite old at this point but so this is a 2d problem this is an axisymmetric problem it is it's really just thriving so the axis of symmetry is here really all we're doing here is trying to drive the peg into a tapered hole okay notice one other thing too that both sides of the center face are deformable right so we said that these grids are going to sort of follow the solid is it deforms so one of the problems here is that I'm going to pick even these nodes or these nodes and say don't interpenetrate the other surface as we do the simulation that surface or that body that that node is looking at is faceted right and it's completely artificial that faceting comes from the fact that I put a finite element grid on that solid right and you can actually see that faceting as you get bumps in the forces right so from a physical point of view if you're sitting there on your computers your coffee and you see this you sort of feel good you think okay well it's at least informing and enforcing the constraints but you know we're usually interested in backing out things like what are the forces that are developed is it going to damage the surface this kind of thing and it's hard to make a lot of sense out of this we certainly know it's not physical right so that gives us maybe at least some kind of clue that this collocation is not the right thing to do I'm giving away part of my answer here but about the time you started to work on this people were thinking about problems of just joining Brits okay which if you think about it I'm going to talk a little bit more about it in a second is a really handy tool to have if you've done any finite element method work just as an analyst right you'll realize you may use a CAD tool to generate a grid fertile piece and you may use a different CAD tool or another description to generate model for another thing and those things are supposed to be glued together it's a real pain in the neck oftentimes to make those grids match up right your finite element textbooks will tell you the grid should match up the convergence characteristics are only guaranteed if the grids are conformed right well who cares if they're not well one reason you'd care remember we said we wanted to be able to guarantee your rate of convergence right so if you're using linear elements you know in other words piecewise linear approximations of displacement for example then the theory tells you because the stretches should converge at order one right as you refine the grid and what happens is when you don't match up grids you don't get that still converges but you can actually see an obvious degradation remembering that we're gonna have really fine scale stuff on the interfaces that we want to resolve this is something if we can fix we really like to this particular example is one where there's sort of a gala you put in here and there was actually friction on it but it's the same idea whether I'm trying to glue that interface together or whether in the applications we're going to consider I want it to be pretty slide bad degradation of convergence is a problem so I would actually take that much joining problem and use it to sort of describe our proposed solution so the Mordor finite element method if you're not used to thinking about numerix you can literally think about it where the name comes from is the order between bricks so the imagine a brick wall the bricks of your elements and the question we're asking here is how do we design that more in a mathematical sense to give us some of these characteristics we said we wanted the right convergence rate and so forth and so on a little bit more robustness so I'm gonna consider the time problem first well you just gotta make a CAD model one CAD model two I want to glue them together forget about all the contact stuff I just want to glue them together okay kinda like and that's that's that's sort of the approach here now by the way I should go back to this if I were to do the collocation approach I would just take each one of these nodes on this interface and say you can't move relative to that surface that you're sitting on so you literally pin it there okay that has the problems we discussed before right you got less convergence patient to get non-physical results so so the question is is there something else you can do and what you end up with you want in a continuum mechanical sense you want the displacements to match on either side of the interface that's the idea of gluing those two two models together so we're going to introduce the LeBrons multiplier and we're going to do that using an area integral okay now for those of you who know a little bit about the Americas if we were going to do a collocation method then there's a whole question of how we going to approximate spatially that multiplier field right what I would do if I were doing a collocation method is it would be a sum over all the surface nodes of Delta functions write times nodal values what you do in a mortar method is you just use the same shape functions you're using further displacement it's kind of a natural choice actually your that's right and you use those to sort of expand if you like the approximation for the multiplier field over the surface and you just plug it into the equations and you crank it out and what you end up with is a set of notice constraints these CAS but now they're little area integrals and they're really tricky area integrals because they're either products of shape functions on one surface which shape functions on the other surface and this is why nobody did it because it was such a nightmare to figure out how to evaluate these integrals so I'm not sort of give you an idea in a second as to why that is now my let me handle contact and I'm just gonna sort of obscure a lot of details in the interest of time if I'm doing the tying problem that I've just got a field once I solve this issue I've just got a field of these force you know it's basically attraction over the over the interface if I want to admit contact then all I need to do is take that traction resolve it into a normal part and attention apart and then impose those constitutive models we talked about before so the out of contact in contact in the normal direction and then something here that if you're familiar with plasticity theory looks a lot like a sweat condition or a yield condition I should say in plasticity so there's no particular point in dwelling on this too much once we figured this out the real point is how we're going to do those approximations of those inner products so integral over the surface of shape function on one surface times a shape function under another surface there's a problem there right I know a little bit about finite elements the underlying expansions describing those shape functions are different right for the two different bodies you have to some kind of resolve that so what we do in 2d is actually not too much trouble you sort of take this surface so here's body one here's body two you parse it up into piecewise smooth pieces and you can sort of go ahead and evaluate those integrals sum over the interface it's actually not too bad in 3d it's a bit of a nightmare so now I'm kind of looking down on the contact surface the red dots are sort of the surface notes of one body the black dots to the surface nodes of another body and you can see pretty quickly that even for low order elements pressing on low order elements those sort of smooth regions which I need do those integrals properly are polygamous okay they can have three five six they can have any number of sides so we actually a few years ago with Mike – Soylent livermore figured out a way they do these integrals and then another detail I'm not going to talk much about here it's one thing to evaluate the integrals but then remember our global system of equations is nonlinear we're going to use newton-raphson iteration of some sort to sort of get this thing to equilibrium so I not only need to evaluate these integrals but I also need to evaluate their derivatives with respect to the underlying displacement fields so it is involved but it is doable and basically the idea is it better be worth it okay so let me kind of recap where we are so I still have yet to show you although I've given you a little bit of evidence that if I don't in this trouble and the folks from the deep domain decomposition community did this work for us a few years ago they showed us that if you go ahead and force these constraints in this sort of way you'll get back those rates of convergence we expect to get from the method well that's a pretty good deal right because we're violating one of the basic rules of finite elements which is we're not worried about the grid being compatible and we can still recover the right rate of convergence but as painful right we have to do all of these silly integrals and linearize and all that but that was our original motivation for doing it because I wanted to be able to tell you as a user you're using this method you're using whatever slip wha you think is appropriate for the system you need to describe there's a there's a rate of convergence to it there's an accuracy and the tractions that you can expect out of it all of that was our primary goal turns out you get one other thing too which we didn't really anticipate and this is that the message is more robust it actually enables you to take bigger load steps it converges a little bit more easily it smooths out a lot of the operators to the point where it actually works a little better I'll give you an example does the think this actually works we'll see so this is a really silly example but it actually gives you the idea I'll run that again in a second the idea here is I'm going to take this red block and push it down into the base block and I'm just going to try to move it tangentially now think about what's gonna happen if I do sort of the old node collocation approach method those nodes kind of fall off the cliffs right so I push it down I develop a very high nodal force to prevent the inner penetration at each one of those nodes but then as I move this thing progressively closer to the edge at some point these noes leave off the edge that force goes to zero and the newton-raphson method just can't converging right the point is that with a mortar which I'm showing you here on the Left it's smooth those operators because the constraints are non-local right they're evaluated over those sort of funny polygonal areas that I showed you before so it smoothes the operators and so there are a lot of problems like this particularly for deformable to perform about contact problems where you sort of can't get these traditional methods to even work okay and and because of the smoothing here so you you know that's the sort of unexpected bonus we got here was the robustness and so this is Ben Yang's work who is one of my PhD students a few years ago the one on the right is dynamic the one on the right I'm sorry the one on the left is dynamic the one on the right which I apologize for the quality of this actually a chrome-plated cylinder okay so so in this particular case this is the kind of thing that folks who are doing crumpled design and for in places like this are interested in you actually an important dissipation energy mechanism is to be able to crumple these things so we like to be able to simulate that contacts happening throughout so so we were able to sort of show that not only does this method gives you more robustness but that it was pretty generally applicable and another Michelin type problem which actually got been his first job because he went to abacus he showed this very movement and they knew that none of their versions of abacus could do this problem and they offer them the job so so it's it's good sometimes if you're graduate student to have that killer app show somebody doesn't look like very much now the idea here is this is actually a run flat tire which was all the rage for a while and if you know anything about run-flat tires you know the failure mechanism is basically on the inside of the tire they put a lubricant in there so that the thing can run flat for some distance but at some point that lubricant gets hot enough that it starts melting the rubber right so so the idea is they just wanted to understand again those dissipation mechanism as well so so it's a pretty cool method right it is not the tool for all profits because it is expensive you know doing all of those area integrals that I showed it's an involved analysis it's much more heavily coupled upon across the interface that in defending traditional methods but oftentimes it's difference between being able to do the problem and not okay because of the robustness and we didn't really guess the robustness that was something we found out at the time so what I when I sort of promised to do in the end is to show how some of these ideas are interesting in terms of extending them to other applications that involve either different types of contact or fluid structure interactions so this was actually work that we were doing for Sandia a few years ago with the College of mine John Davao he works on the extended finite element method but what the Sandia guys wanted to do is look at crystalline micro structures and what they were curious of or curious about is is there a way to take a topology like this and this is 2d so 3d is much harder again you do what your finite element textbook tells you you need to do your grid should be compatible of the cross grain boundaries right that's maybe impractical in 2d and it's almost impossible in 3d and just to give you another level of complication for this these guys wanted to do stochastic methods so they wanted to be able to do these simulations thousands of times and sort of a Monte Carlo sense so they really didn't want to do this they even want to have to generate these grids that over each grain boundary match done what they were interested in is can I give it rid like this which pays no respect whatsoever to grain boundaries sort of what extended finite element methods are designed to do and give me a reasonable answer okay and I'm gonna go a little bit quickly here so I don't run out of time but we were also involved with some energy harvesting work which leads to the FSI stuff I'll show at the end but how this interest is in this as well so the idea here is how do we take now let me give you just an idea of the equation so suppose I've got two grains I've told you now that I'm gonna try to figure out a way that I don't have to put element boundaries exactly on that interface that they can try to pass through somewhat arbitrarily so really what we do here you do what the x-men people call enriched the displacement field okay and what this really means and I think I've got a picture here that shows this pretty well is you build in the possibility for jumps in the displacement field as you cross the grain boundary okay so this is not a find an element method anymore it's it's what we would call an enriched or an enhanced or an extended finite element method and you use these sort of spatial Heaviside functions to do it so skipping a lot of details what we're going to be after here is using methods like this to allow these grain boundaries if you like to come apart and then we're going to put the LaBranche multipliers back in remember the time problem right to sort of enforce compatibility or if we want to let it slide we can let it slide okay so so you're sort of first naive impression would be well let's use the trick that worked when we were doing contact problems which is we just used the displacement shape functions to describe those multipliers that is a disaster and I'm going to skip the equations and just show you what you get so this is a simple beam bending problem this is a numerical oscillation right so these are literally multipliers on this interface as you move along it and basically what we would say here is this is not a stable approximation okay it turns out though that it's pretty easy to fix and the idea here behind stabilizing this multiplier field is simply to put in more displacement degrees of freedom or what are call couples okay and if you studied finite element methods a little bit in fact this comes up and needed the press ability as well you have something called the LDP condition and basically that condition says that when you have a mixed problem in this case between pressures and displacements there's sort of an inequality that needs to be obeyed between the number of displacement degrees of freedom roughly speaking and the number of pressure degrees of freedom and if that gets too different from what's happening in the underlying medium you get the sort of oscillations we saw a couple of slides ago so this was a pretty standard stabilization method and it actually works and it also gives you the proper rate of convergence and so what people have actually been doing for quite some time now is using these sort of they're related to martyr methods but you have to enrich the displacement field to actually give the analyst flexibility to not have to grit all these page break examples so this is a pretty attractive method and again you see the unstable result on the upper left and then you see the state result on the upper-right and finally I'll just sort of conclude with fluid structure interaction similar sort of motivation different expound but the way you want to think about this here is suppose I've got I don't know think of a bathtub I've got a deforming squishy thing that I'm going to use a little grungy and solid for back to me this blob be floating enough in a pool of water okay and the idea here is I'd like to use an oil areand grid for the water kind of like I was talking about the beginning with a tire going through a puddle and I want to use a Lagrangian grid here that's going to give me a compatible grids on that fluid structure interaction examples to see the effectiveness of the method Jessica Sanders who actually did this work with me and then ended up going out to Livermore to work with with my clue selling some of their production codes they rated the show was both fluid mechanics and solid mechanics applications that the same idea is a stabilization and not only worked quite well that they can be generalized to large deformations as well and they're really discontinuous galerkin methods which I won't get into here but but it's really the same principle that we've been discussing in that that makes those go so just to sort of summarize my story and pretty well done we've got take any questions probably not surprisingly it's oftentimes this true in science but we sort of look back and go well that should have been obvious right so finite element methods are predicated on the idea of taking the governing equations and weighting them in a variational way over volumes of material right it probably shouldn't be a surprise that we should do the same thing with interface mechanics right which is effectively what we're doing okay but sort of the nice surprise it's the right thing to do give you the convergence you should get bla bla bla but it also really has this nice benefit in large deformation problems of giving you quite a bit more robustness if you have her have the experience as an analyst of running a non linear finite element code and doing contact you'll know how frustrating this because this sort of non smoothness bites you all the time right so so it the the smoothing that these operators give you is really attractive and then the other thing we've learned and that I tried to show rather quickly at the end is if you extend these to expound and embedded interfaces you can do it but you can't do it in a naive way there has to be a notion of stabilization there and I think this is really an active area at work now there are a lot of folks that are really thinking about FSI in a different way and the methods that they're using to do that at least in the finite element world are quite closely related [Applause] but ended contacting a better excuse yeah so I was cool on to when assimilation doesn't work and I wonder is resolving a flurry for problem right better than I thought right so yeah he's right so he's asking the question if the original continuum mechanical problem is not well posed maybe one way of interpreting your question is how is that going to show up here and how will we know that if we have a problem you know it's so it's pretty terroristic I sort of have to admit that the most serious well posedness problems are actually in Coulomb friction and if you go back to the work that Kikuchi and Odin did and 80s they they give you a number of examples of you know actually to be rigorously correct to even take a Coulomb friction law which by the way never works but we insist as engineers and using that right so physics be damned we still use : it also has mathematical problems and they're pretty well documented so that chance is already is always out there right the kind of numerical difficulties though that I'm talking about here and are not one there they're ones where if we apply a disk Rinna's ation method or a different scheme or whatever we can get an answer okay the case is where I was showing convergence there is a known solution that we're sort of evaluating convergence to so in that sense it's a little bit rigged these these lack of uniqueness issues and that's really what tends to come up quite often with Coulomb friction though are recognized we don't actually see them too much in practice but it's always a possibility and there are pathological cases I think just to give you one which I think you know from your own work is if you're talking about dynamical systems particularly that have the opportunity to bifurcate then then it becomes very difficult to know whether the difficulties that you're seeing are actually something that's real something that's in the governing equations or something that is introduced with your your approximation so it's always there but there at least the applications I've showed here there's reasonable evidence to suggest that the problems are are well posed at least in fact we did it before we didn't work and we were trying and our motivation purely was to spatially smooth the operators and it actually works but it still has the accuracy problems and in fact in effect it's even worse because it's hard to prove theorems when you when you sort of take the finite element discretization and you say I'm going to approximate that and the contact operator with a spline right if you can do it and and actually in terms of robustness it works quite well I did some work with Mike Puzo two or three years prior to this and that's exactly what we did but you can't really probe prove theorems about it so so you don't you don't typically know how much accuracy you're getting that's the downfall but you know if you look at most of the production level codes that are out there it's got contacts and it'll begin is a popular method so the physical problem often won't be still this contact is actually feels good getting inside work because coming out of the applications have situations we have and I don't think we've solved them very well I mean that the work that I've seen that's probably closest to this is folks who are looking at these sort of hybrid approaches where you take out of mystic simulations on how and you couple it in we we haven't done that ourselves I I think there probably is some wisdom buried in this about how that coupling could be done you know spatially to be numerically robots it's a good idea is but it's not something we've really had the chance at least in pursuit but yeah the whole multiscale I mean I think they're sort of there's there's dissipation and there's at least the potential over time but even in steady state rolling as I mentioned sort of Italian there is that highly localized region where slippers and they call it a flea and it's very hard to resolve and it really you would want a multiscale not right to do it but then I think the the the other thing is actually kind of interesting to think about entire designers do this all the time is there's there's always trade-offs between say the geometry of the sights and grooves and things like noise money so if you're trying to minimize noise that's going to give you a different tread design then say is going to give you dissipation inside the tire below a certain level you see when getting home so so they're they're having to make compromises like this all the time it's complicated but but that would be sort of my first order answer the other one of course and maybe this is more applicable to truck tires than than passenger tires but you just have to look at the carcasses by that by the freeway to realize that there's stuff going on inside right between that sort of sculpture of the tire and the fans underneath and there's dissipation that occurs there too and in truck tires that delamination usually why we see those pieces in fire by the side of the road so there's a few things I'm sorry I didn't quite get how did I simulate which in the run fockin okay yeah not really the degradation just the dissipation right so again this is sort of one of these heavily heuristic areas but if you were to take sort of the product of the frictional stress and the slip velocity that gives you a dissipation right and then if you integrate that over and tire or over a surface that gives you an idea in any instant in time how much thermal energy is being generated you know so that was basically the calculation we were doing the next obvious step which I think maybe is the question you're asking you might want to put a damage model in there or something we weren't trying to model the damage we were just trying to quantify what that dissipation looked like and where it was happening does that make sense I mean I think we're sorted to the point where he couldn't it could be credibly done right I mean think about it this way for those of you who model things like damage you know when we entered this field 30 or 40 years ago all you had was the sets of discrete LeBrons multipliers on an interface and you know you know where do you even start and coming up with an approximation of damage models need to involve things like slip velocity and pressure and frictional traction in history variables and there just wasn't even a kinematic framework to think about this kind of thing so I think now that we have some of that a little bit more established and it turns out to be the right thing to do to America there's a little more hope right you hit that hold two mechanisms because there are multiple mechanisms right and it's just nothing right any other question okay thank you [Applause]